@c DO NOT EDIT!  Generated automatically by munge-texi.

@c Copyright (C) 1996, 1997 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Statistics
@chapter Statistics

I hope that someday Octave will include more statistics functions.  If
you would like to help improve Octave in this area, please contact
@email{bug@@octave.org}.

@menu
* Basic Statistical Functions::  
* Tests::                       
* Models::                      
* Distributions::               
@end menu

@node Basic Statistical Functions
@section Basic Statistical Functions

@anchor{doc-mean}
@deftypefn {Function File} {} mean (@var{x}, @var{dim}, @var{opt})
If @var{x} is a vector, compute the mean of the elements of @var{x}
@iftex
@tex
$$ {\rm mean}(x) = \bar{x} = {1\over N} \sum_{i=1}^N x_i $$
@end tex
@end iftex
@ifinfo

@example
mean (x) = SUM_i x(i) / N
@end example
@end ifinfo
If @var{x} is a matrix, compute the mean for each column and return them
in a row vector.

With the optional argument @var{opt}, the kind of mean computed can be
selected.  The following options are recognized:

@table @code
@item "a"
Compute the (ordinary) arithmetic mean.  This is the default.

@item "g"
Computer the geometric mean.

@item "h"
Compute the harmonic mean.
@end table

If the optional argument @var{dim} is supplied, work along dimension
@var{dim}.

Both @var{dim} and @var{opt} are optional.  If both are supplied,
either may appear first.
@end deftypefn


@anchor{doc-median}
@deftypefn {Function File} {} median (@var{x})
If @var{x} is a vector, compute the median value of the elements of
@var{x}.
@iftex
@tex
$$
{\rm median} (x) =
  \cases{x(\lceil N/2\rceil), & $N$ odd;\cr
          (x(N/2)+x(N/2+1))/2, & $N$ even.}
$$
@end tex
@end iftex
@ifinfo

@example
@group
            x(ceil(N/2)),             N odd
median(x) =
            (x(N/2) + x((N/2)+1))/2,  N even
@end group
@end example
@end ifinfo
If @var{x} is a matrix, compute the median value for each
column and return them in a row vector.
@end deftypefn

@seealso{std and mean}


@anchor{doc-std}
@deftypefn {Function File} {} std (@var{x})
@deftypefnx {Function File} {} std (@var{x}, @var{opt})
@deftypefnx {Function File} {} std (@var{x}, @var{opt}, @var{dim})
If @var{x} is a vector, compute the standard deviation of the elements
of @var{x}.
@iftex
@tex
$$
{\rm std} (x) = \sigma (x) = \sqrt{{\sum_{i=1}^N (x_i - \bar{x}) \over N - 1}}
$$
@end tex
@end iftex
@ifinfo

@example
@group
std (x) = sqrt (sumsq (x - mean (x)) / (n - 1))
@end group
@end example
@end ifinfo
If @var{x} is a matrix, compute the standard deviation for
each column and return them in a row vector.

The argument @var{opt} determines the type of normalization to use. Valid values
are

@table @asis 
@item 0:
  normalizes with N-1, provides the square root of best unbiased estimator of 
  the variance [default]
@item 1:
  normalizes with N, this provides the square root of the second moment around 
  the mean
@end table

The third argument @var{dim} determines the dimension along which the standard
deviation is calculated.
@end deftypefn

@seealso{mean and median}


@anchor{doc-cov}
@deftypefn {Function File} {} cov (@var{x}, @var{y})
If each row of @var{x} and @var{y} is an observation and each column is
a variable, the (@var{i}, @var{j})-th entry of
@code{cov (@var{x}, @var{y})} is the covariance between the @var{i}-th
variable in @var{x} and the @var{j}-th variable in @var{y}.  If called
with one argument, compute @code{cov (@var{x}, @var{x})}.
@end deftypefn


@anchor{doc-corrcoef}
@deftypefn {Function File} {} corrcoef (@var{x}, @var{y})
If each row of @var{x} and @var{y} is an observation and each column is
a variable, the (@var{i}, @var{j})-th entry of
@code{corrcoef (@var{x}, @var{y})} is the correlation between the
@var{i}-th variable in @var{x} and the @var{j}-th variable in @var{y}.
If called with one argument, compute @code{corrcoef (@var{x}, @var{x})}.
@end deftypefn


@anchor{doc-kurtosis}
@deftypefn {Function File} {} kurtosis (@var{x}, @var{dim})
If @var{x} is a vector of length @math{N}, return the kurtosis
@iftex
@tex
$$
 {\rm kurtosis} (x) = {1\over N \sigma(x)^4} \sum_{i=1}^N (x_i-\bar{x})^4 - 3
$$
@end tex
@end iftex
@ifinfo

@example
kurtosis (x) = N^(-1) std(x)^(-4) sum ((x - mean(x)).^4) - 3
@end example
@end ifinfo

@noindent
of @var{x}.  If @var{x} is a matrix, return the kurtosis over the
first non-singleton dimension. The optional argument @var{dim}
can be given to force the kurtosis to be given over that 
dimension.
@end deftypefn


@anchor{doc-mahalanobis}
@deftypefn {Function File} {} mahalanobis (@var{x}, @var{y})
Return the Mahalanobis' D-square distance between the multivariate
samples @var{x} and @var{y}, which must have the same number of
components (columns), but may have a different number of observations
(rows).
@end deftypefn


@anchor{doc-skewness}
@deftypefn {Function File} {} skewness (@var{x}, @var{dim})
If @var{x} is a vector of length @math{n}, return the skewness
@iftex
@tex
$$
{\rm skewness} (x) = {1\over N \sigma(x)^3} \sum_{i=1}^N (x_i-\bar{x})^3
$$
@end tex
@end iftex
@ifinfo

@example
skewness (x) = N^(-1) std(x)^(-3) sum ((x - mean(x)).^3)
@end example
@end ifinfo

@noindent
of @var{x}.  If @var{x} is a matrix, return the skewness along the
first non-singleton dimension of the matrix. If the optional
@var{dim} argument is given, operate along this dimension.
@end deftypefn


@c XXX FIXME XXX -- these need to be organized.

@anchor{doc-values}
@deftypefn {Function File} {} values (@var{x})
Return the different values in a column vector, arranged in ascending
order.
@end deftypefn


@anchor{doc-var}
@deftypefn {Function File} {} var (@var{x})
For vector arguments, return the (real) variance of the values.
For matrix arguments, return a row vector contaning the variance for
each column.

The argument @var{opt} determines the type of normalization to use. Valid 
values are

@table @asis 
@item 0:
  normalizes with N-1, provides the square root of best unbiased estimator
  of the variance [default]
@item 1:
  normalizes with N, this provides the square root of the second moment
  around the mean
@end table

The third argument @var{dim} determines the dimension along which the 
variance is calculated.
@end deftypefn


@anchor{doc-table}
@deftypefn {Function File} {[@var{t}, @var{l_x}] =} table (@var{x})
@deftypefnx {Function File} {[@var{t}, @var{l_x}, @var{l_y}] =} table (@var{x}, @var{y})
Create a contingency table @var{t} from data vectors.  The @var{l}
vectors are the corresponding levels.

Currently, only 1- and 2-dimensional tables are supported.
@end deftypefn


@anchor{doc-studentize}
@deftypefn {Function File} {} studentize (@var{x}, @var{dim})
If @var{x} is a vector, subtract its mean and divide by its standard
deviation.

If @var{x} is a matrix, do the above along the first non-singleton
dimension. If the optional argument @var{dim} is given then operate
along this dimension.
@end deftypefn


@anchor{doc-statistics}
@deftypefn {Function File} {} statistics (@var{x})
If @var{x} is a matrix, return a matrix with the minimum, first
quartile, median, third quartile, maximum, mean, standard deviation,
skewness and kurtosis of the columns of @var{x} as its rows.

If @var{x} is a vector, treat it as a column vector.
@end deftypefn


@anchor{doc-spearman}
@deftypefn {Function File} {} spearman (@var{x}, @var{y})
Compute Spearman's rank correlation coefficient @var{rho} for each of
the variables specified by the input arguments.

For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.

@code{spearman (@var{x})} is equivalent to @code{spearman (@var{x},
@var{x})}.

For two data vectors @var{x} and @var{y}, Spearman's @var{rho} is the
correlation of the ranks of @var{x} and @var{y}.

If @var{x} and @var{y} are drawn from independent distributions,
@var{rho} has zero mean and variance @code{1 / (n - 1)}, and is
asymptotically normally distributed.
@end deftypefn


@anchor{doc-run_count}
@deftypefn {Function File} {} run_count (@var{x}, @var{n})
Count the upward runs along the first non-singleton dimension of
@var{x} of length 1, 2, ..., @var{n}-1 and greater than or equal 
to @var{n}. If the optional argument @var{dim} is given operate
along this dimension
@end deftypefn


@anchor{doc-ranks}
@deftypefn {Function File} {} ranks (@var{x}, @var{dim})
If @var{x} is a vector, return the (column) vector of ranks of
@var{x} adjusted for ties.

If @var{x} is a matrix, do the above for along the first 
non-singleton dimension. If the optional argument @var{dim} is
given, operate along this dimension.
@end deftypefn


@anchor{doc-range}
@deftypefn {Function File} {} range (@var{x})
@deftypefnx {Function File} {} range (@var{x}, @var{dim})
If @var{x} is a vector, return the range, i.e., the difference
between the maximum and the minimum, of the input data.

If @var{x} is a matrix, do the above for each column of @var{x}.

If the optional argument @var{dim} is supplied, work along dimension
@var{dim}.
@end deftypefn


@anchor{doc-qqplot}
@deftypefn {Function File} {[@var{q}, @var{s}] =} qqplot (@var{x}, @var{dist}, @var{params})
Perform a QQ-plot (quantile plot).

If F is the CDF of the distribution @var{dist} with parameters
@var{params} and G its inverse, and @var{x} a sample vector of length
@var{n}, the QQ-plot graphs ordinate @var{s}(@var{i}) = @var{i}-th
largest element of x versus abscissa @var{q}(@var{i}f) = G((@var{i} -
0.5)/@var{n}).

If the sample comes from F except for a transformation of location
and scale, the pairs will approximately follow a straight line.

The default for @var{dist} is the standard normal distribution.  The
optional argument @var{params} contains a list of parameters of
@var{dist}.  For example, for a quantile plot of the uniform
distribution on [2,4] and @var{x}, use

@example
qqplot (x, "uniform", 2, 4)
@end example

If no output arguments are given, the data are plotted directly.
@end deftypefn


@anchor{doc-probit}
@deftypefn {Function File} {} probit (@var{p})
For each component of @var{p}, return the probit (the quantile of the
standard normal distribution) of @var{p}.
@end deftypefn


@anchor{doc-ppplot}
@deftypefn {Function File} {[@var{p}, @var{y}] =} ppplot (@var{x}, @var{dist}, @var{params})
Perform a PP-plot (probability plot).

If F is the CDF of the distribution @var{dist} with parameters
@var{params} and @var{x} a sample vector of length @var{n}, the
PP-plot graphs ordinate @var{y}(@var{i}) = F (@var{i}-th largest
element of @var{x}) versus abscissa @var{p}(@var{i}) = (@var{i} -
0.5)/@var{n}.  If the sample comes from F, the pairs will
approximately follow a straight line.

The default for @var{dist} is the standard normal distribution.  The
optional argument @var{params} contains a list of parameters of
@var{dist}.  For example, for a probability plot of the uniform
distribution on [2,4] and @var{x}, use

@example
ppplot (x, "uniform", 2, 4)
@end example

If no output arguments are given, the data are plotted directly.
@end deftypefn


@anchor{doc-moment}
@deftypefn {Function File} {} moment (@var{x}, @var{p}, @var{opt}, @var{dim})
If @var{x} is a vector, compute the @var{p}-th moment of @var{x}.

If @var{x} is a matrix, return the row vector containing the
@var{p}-th moment of each column.

With the optional string opt, the kind of moment to be computed can
be specified.  If opt contains @code{"c"} or @code{"a"}, central
and/or absolute moments are returned.  For example,

@example
moment (x, 3, "ac")
@end example

@noindent
computes the third central absolute moment of @var{x}.

If the optional argument @var{dim} is supplied, work along dimension
@var{dim}.
@end deftypefn


@anchor{doc-meansq}
@deftypefn {Function File} {} meansq (@var{x})
@deftypefnx {Function File} {} meansq (@var{x}, @var{dim})
For vector arguments, return the mean square of the values.
For matrix arguments, return a row vector contaning the mean square
of each column. With the optional @var{dim} argument, returns the
mean squared of the values along this dimension
@end deftypefn


@anchor{doc-logit}
@deftypefn {Function File} {} logit (@var{p})
For each component of @var{p}, return the logit @code{log (@var{p} /
(1-@var{p}))} of @var{p}.
@end deftypefn


@anchor{doc-kendall}
@deftypefn {Function File} {} kendall (@var{x}, @var{y})
Compute Kendall's @var{tau} for each of the variables specified by
the input arguments.

For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.

@code{kendall (@var{x})} is equivalent to @code{kendall (@var{x},
@var{x})}.

For two data vectors @var{x}, @var{y} of common length @var{n},
Kendall's @var{tau} is the correlation of the signs of all rank
differences of @var{x} and @var{y};  i.e., if both @var{x} and
@var{y} have distinct entries, then

@iftex
@tex
$$ \tau = {1 \over n(n-1)} \sum_{i,j} {\rm sign}(q_i-q_j) {\rm sign}(r_i-r_j) $$
@end tex
@end iftex
@ifinfo
@example
         1    
tau = -------   SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
      n (n-1)   i,j
@end example
@end ifinfo

@noindent
in which the
@iftex
@tex
$q_i$ and $r_i$
@end tex
@end iftex
@ifinfo
@var{q}(@var{i}) and @var{r}(@var{i})
@end ifinfo
 are the ranks of
@var{x} and @var{y}, respectively.

If @var{x} and @var{y} are drawn from independent distributions,
Kendall's @var{tau} is asymptotically normal with mean 0 and variance
@code{(2 * (2@var{n}+5)) / (9 * @var{n} * (@var{n}-1))}.
@end deftypefn


@anchor{doc-iqr}
@deftypefn {Function File} {} iqr (@var{x}, @var{dim})
If @var{x} is a vector, return the interquartile range, i.e., the
difference between the upper and lower quartile, of the input data.

If @var{x} is a matrix, do the above for first non singleton
dimension of @var{x}.. If the option @var{dim} argument is given,
then operate along this dimension.
@end deftypefn


@anchor{doc-cut}
@deftypefn {Function File} {} cut (@var{x}, @var{breaks})
Create categorical data out of numerical or continuous data by
cutting into intervals.

If @var{breaks} is a scalar, the data is cut into that many
equal-width intervals.  If @var{breaks} is a vector of break points,
the category has @code{length (@var{breaks}) - 1} groups.

The returned value is a vector of the same size as @var{x} telling
which group each point in @var{x} belongs to.  Groups are labelled
from 1 to the number of groups; points outside the range of
@var{breaks} are labelled by @code{NaN}.
@end deftypefn


@anchor{doc-cor}
@deftypefn {Function File} {} cor (@var{x}, @var{y})
The (@var{i}, @var{j})-th entry of @code{cor (@var{x}, @var{y})} is
the correlation between the @var{i}-th variable in @var{x} and the
@var{j}-th variable in @var{y}.

For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.

@code{cor (@var{x})} is equivalent to @code{cor (@var{x}, @var{x})}.
@end deftypefn


@anchor{doc-cloglog}
@deftypefn {Function File} {} cloglog (@var{x})
Return the complementary log-log function of @var{x}, defined as

@example
- log (- log (@var{x}))
@end example
@end deftypefn


@anchor{doc-center}
@deftypefn {Function File} {} center (@var{x})
@deftypefnx {Function File} {} center (@var{x}, @var{dim})
If @var{x} is a vector, subtract its mean.
If @var{x} is a matrix, do the above for each column.
If the optional argument @var{dim} is given, perform the above
operation along this dimension
@end deftypefn


@node Tests
@section Tests

@anchor{doc-anova}
@deftypefn {Function File} {[@var{pval}, @var{f}, @var{df_b}, @var{df_w}] =} anova (@var{y}, @var{g})
Perform a one-way analysis of variance (ANOVA).  The goal is to test
whether the population means of data taken from @var{k} different
groups are all equal.

Data may be given in a single vector @var{y} with groups specified by
a corresponding vector of group labels @var{g} (e.g., numbers from 1
to @var{k}). This is the general form which does not impose any
restriction on the number of data in each group or the group labels.

If @var{y} is a matrix and @var{g} is omitted, each column of @var{y}
is treated as a group.  This form is only appropriate for balanced
ANOVA in which the numbers of samples from each group are all equal.

Under the null of constant means, the statistic @var{f} follows an F
distribution with @var{df_b} and @var{df_w} degrees of freedom.

The p-value (1 minus the CDF of this distribution at @var{f}) is
returned in @var{pval}.

If no output argument is given, the standard one-way ANOVA table is
printed.
@end deftypefn


@anchor{doc-bartlett_test}
@deftypefn {Function File} {[@var{pval}, @var{chisq}, @var{df}] =} bartlett_test (@var{x1}, @dots{}) 
Perform a Bartlett test for the homogeneity of variances in the data
vectors @var{x1}, @var{x2}, @dots{}, @var{xk}, where @var{k} > 1.

Under the null of equal variances, the test statistic @var{chisq}
approximately ollows a chi-square distribution with @var{df} degrees of
freedom.

The p-value (1 minus the CDF of this distribution at @var{chisq}) is
returned in @var{pval}.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-chisquare_test_homogeneity}
@deftypefn {Function File} {[@var{pval}, @var{chisq}, @var{df}] =} chisquare_test_homogeneity (@var{x}, @var{y}, @var{c})
Given two samples @var{x} and @var{y}, perform a chisquare test for
homogeneity of the null hypothesis that @var{x} and @var{y} come from
the same distribution, based on the partition induced by the
(strictly increasing) entries of @var{c}.

For large samples, the test statistic @var{chisq} approximately follows a
chisquare distribution with @var{df} = @code{length (@var{c})}
degrees of freedom.

The p-value (1 minus the CDF of this distribution at @var{chisq}) is
returned in @var{pval}.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-chisquare_test_independence}
@deftypefn {Function File} {[@var{pval}, @var{chisq}, @var{df}] =} chisquare_test_independence (@var{x})
Perform a chi-square test for indepence based on the contingency
table @var{x}.  Under the null hypothesis of independence,
@var{chisq} approximately has a chi-square distribution with
@var{df} degrees of freedom.

The p-value (1 minus the CDF of this distribution at chisq) of the
test is returned in @var{pval}.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-cor_test}
@deftypefn {Function File} {} cor_test (@var{x}, @var{y}, @var{alt}, @var{method})
Test whether two samples @var{x} and @var{y} come from uncorrelated
populations.

The optional argument string @var{alt} describes the alternative
hypothesis, and can be @code{"!="} or @code{"<>"} (non-zero),
@code{">"} (greater than 0), or @code{"<"} (less than 0).  The
default is the two-sided case.

The optional argument string @var{method} specifies on which
correlation coefficient the test should be based.  If @var{method} is
@code{"pearson"} (default), the (usual) Pearson's product moment
correlation coefficient is used.  In this case, the data should come
from a bivariate normal distribution.  Otherwise, the other two
methods offer nonparametric alternatives. If @var{method} is
@code{"kendall"}, then Kendall's rank correlation tau is used.  If
@var{method} is @code{"spearman"}, then Spearman's rank correlation
rho is used.  Only the first character is necessary.

The output is a structure with the following elements:

@table @var
@item pval
The p-value of the test.
@item stat
The value of the test statistic.
@item dist
The distribution of the test statistic.
@item params
The parameters of the null distribution of the test statistic.
@item alternative
The alternative hypothesis.
@item method
The method used for testing.
@end table

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-f_test_regression}
@deftypefn {Function File} {[@var{pval}, @var{f}, @var{df_num}, @var{df_den}] =} f_test_regression (@var{y}, @var{x}, @var{rr}, @var{r})
Perform an F test for the null hypothesis rr * b = r in a classical
normal regression model y = X * b + e.

Under the null, the test statistic @var{f} follows an F distribution
with @var{df_num} and @var{df_den} degrees of freedom.

The p-value (1 minus the CDF of this distribution at @var{f}) is
returned in @var{pval}.

If not given explicitly, @var{r} = 0.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-hotelling_test}
@deftypefn {Function File} {[@var{pval}, @var{tsq}] =} hotelling_test (@var{x}, @var{m})
For a sample @var{x} from a multivariate normal distribution with unknown
mean and covariance matrix, test the null hypothesis that @code{mean
(@var{x}) == @var{m}}.

Hotelling's T^2 is returned in @var{tsq}.  Under the null,
@math{(n-p) T^2 / (p(n-1))} has an F distribution with @math{p} and
@math{n-p} degrees of freedom, where @math{n} and @math{p} are the
numbers of samples and variables, respectively.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-hotelling_test_2}
@deftypefn {Function File} {[@var{pval}, @var{tsq}] =} hotelling_test_2 (@var{x}, @var{y})
For two samples @var{x} from multivariate normal distributions with
the same number of variables (columns), unknown means and unknown
equal covariance matrices, test the null hypothesis @code{mean
(@var{x}) == mean (@var{y})}.

Hotelling's two-sample T^2 is returned in @var{tsq}.  Under the null,

@example
(n_x+n_y-p-1) T^2 / (p(n_x+n_y-2))
@end example

@noindent
has an F distribution with @math{p} and @math{n_x+n_y-p-1} degrees of
freedom, where @math{n_x} and @math{n_y} are the sample sizes and
@math{p} is the number of variables.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-kolmogorov_smirnov_test}
@deftypefn {Function File} {[@var{pval}, @var{ks}] =} kolmogorov_smirnov_test (@var{x}, @var{dist}, @var{params}, @var{alt})
Perform a Kolmogorov-Smirnov test of the null hypothesis that the
sample @var{x} comes from the (continuous) distribution dist. I.e.,
if F and G are the CDFs corresponding to the sample and dist,
respectively, then the null is that F == G.

The optional argument @var{params} contains a list of parameters of
@var{dist}.  For example, to test whether a sample @var{x} comes from
a uniform distribution on [2,4], use

@example
kolmogorov_smirnov_test(x, "uniform", 2, 4)
@end example

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative F
!= G.  In this case, the test statistic @var{ks} follows a two-sided
Kolmogorov-Smirnov distribution.  If @var{alt} is @code{">"}, the
one-sided alternative F > G is considered.  Similarly for @code{"<"},
the one-sided alternative F > G is considered.  In this case, the
test statistic @var{ks} has a one-sided Kolmogorov-Smirnov
distribution.  The default is the two-sided case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-kolmogorov_smirnov_test_2}
@deftypefn {Function File} {[@var{pval}, @var{ks}, @var{d}] =} kolmogorov_smirnov_test_2 (@var{x}, @var{y}, @var{alt})
Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis
that the samples @var{x} and @var{y} come from the same (continuous)
distribution.  I.e., if F and G are the CDFs corresponding to the
@var{x} and @var{y} samples, respectively, then the null is that F ==
G.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative F
!= G.  In this case, the test statistic @var{ks} follows a two-sided
Kolmogorov-Smirnov distribution.  If @var{alt} is @code{">"}, the
one-sided alternative F > G is considered.  Similarly for @code{"<"},
the one-sided alternative F < G is considered.  In this case, the
test statistic @var{ks} has a one-sided Kolmogorov-Smirnov
distribution.  The default is the two-sided case.

The p-value of the test is returned in @var{pval}.

The third returned value, @var{d}, is the test statistic, the maximum
vertical distance between the two cumulative distribution functions.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-kruskal_wallis_test}
@deftypefn {Function File} {[@var{pval}, @var{k}, @var{df}] =} kruskal_wallis_test (@var{x1}, @dots{})
Perform a Kruskal-Wallis one-factor "analysis of variance".

Suppose a variable is observed for @var{k} > 1 different groups, and
let @var{x1}, @dots{}, @var{xk} be the corresponding data vectors.

Under the null hypothesis that the ranks in the pooled sample are not
affected by the group memberships, the test statistic @var{k} is
approximately chi-square with @var{df} = @var{k} - 1 degrees of
freedom.

The p-value (1 minus the CDF of this distribution at @var{k}) is
returned in @var{pval}.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-manova}
@deftypefn {Function File} {} manova (@var{y}, @var{g})
Perform a one-way multivariate analysis of variance (MANOVA). The
goal is to test whether the p-dimensional population means of data
taken from @var{k} different groups are all equal.  All data are
assumed drawn independently from p-dimensional normal distributions
with the same covariance matrix.

The data matrix is given by @var{y}.  As usual, rows are observations
and columns are variables.  The vector @var{g} specifies the
corresponding group labels (e.g., numbers from 1 to @var{k}).

The LR test statistic (Wilks' Lambda) and approximate p-values are
computed and displayed.
@end deftypefn


@anchor{doc-mcnemar_test}
@deftypefn {Function File} {[@var{pval}, @var{chisq}, @var{df}] =} mcnemar_test (@var{x})
For a square contingency table @var{x} of data cross-classified on
the row and column variables, McNemar's test can be used for testing
the null hypothesis of symmetry of the classification probabilities.

Under the null, @var{chisq} is approximately distributed as chisquare
with @var{df} degrees of freedom.

The p-value (1 minus the CDF of this distribution at @var{chisq}) is
returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-prop_test_2}
@deftypefn {Function File} {[@var{pval}, @var{z}] =} prop_test_2 (@var{x1}, @var{n1}, @var{x2}, @var{n2}, @var{alt})
If @var{x1} and @var{n1} are the counts of successes and trials in
one sample, and @var{x2} and @var{n2} those in a second one, test the
null hypothesis that the success probabilities @var{p1} and @var{p2}
are the same.  Under the null, the test statistic @var{z}
approximately follows a standard normal distribution.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@var{p1} != @var{p2}.  If @var{alt} is @code{">"}, the one-sided
alternative @var{p1} > @var{p2} is used.  Similarly for @code{"<"},
the one-sided alternative @var{p1} < @var{p2} is used.
The default is the two-sided case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-run_test}
@deftypefn {Function File} {[@var{pval}, @var{chisq}] =} run_test (@var{x})
Perform a chi-square test with 6 degrees of freedom based on the
upward runs in the columns of @var{x}.  Can be used to test whether
@var{x} contains independent data.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value is displayed.
@end deftypefn


@anchor{doc-sign_test}
@deftypefn {Function File} {[@var{pval}, @var{b}, @var{n}] =} sign_test (@var{x}, @var{y}, @var{alt})
For two matched-pair samples @var{x} and @var{y}, perform a sign test
of the null hypothesis PROB (@var{x} > @var{y}) == PROB (@var{x} <
@var{y}) == 1/2.  Under the null, the test statistic @var{b} roughly
follows a binomial distribution with parameters @code{@var{n} = sum
(@var{x} != @var{y})} and @var{p} = 1/2.

With the optional argument @code{alt}, the alternative of interest
can be selected.  If @var{alt} is @code{"!="} or @code{"<>"}, the
null hypothesis is tested against the two-sided alternative PROB
(@var{x} < @var{y}) != 1/2.  If @var{alt} is @code{">"}, the
one-sided alternative PROB (@var{x} > @var{y}) > 1/2 ("x is
stochastically greater than y") is considered.  Similarly for
@code{"<"}, the one-sided alternative PROB (@var{x} > @var{y}) < 1/2
("x is stochastically less than y") is considered.  The default is
the two-sided case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-t_test}
@deftypefn {Function File} {[@var{pval}, @var{t}, @var{df}] =} t_test (@var{x}, @var{m}, @var{alt})
For a sample @var{x} from a normal distribution with unknown mean and
variance, perform a t-test of the null hypothesis @code{mean
(@var{x}) == @var{m}}.  Under the null, the test statistic @var{t}
follows a Student distribution with @code{@var{df} = length (@var{x})
- 1} degrees of freedom.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{mean (@var{x}) != @var{m}}.  If @var{alt} is @code{">"}, the
one-sided alternative @code{mean (@var{x}) > @var{m}} is considered.
Similarly for @var{"<"}, the one-sided alternative @code{mean
(@var{x}) < @var{m}} is considered,  The default is the two-sided
case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-t_test_2}
@deftypefn {Function File} {[@var{pval}, @var{t}, @var{df}] =} t_test_2 (@var{x}, @var{y}, @var{alt})
For two samples x and y from normal distributions with unknown means
and unknown equal variances, perform a two-sample t-test of the null
hypothesis of equal means.  Under the null, the test statistic
@var{t} follows a Student distribution with @var{df} degrees of
freedom.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{mean (@var{x}) != mean (@var{y})}.  If @var{alt} is @code{">"},
the one-sided alternative @code{mean (@var{x}) > mean (@var{y})} is
used.  Similarly for @code{"<"}, the one-sided alternative @code{mean
(@var{x}) < mean (@var{y})} is used.  The default is the two-sided
case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-t_test_regression}
@deftypefn {Function File} {[@var{pval}, @var{t}, @var{df}] =} t_test_regression (@var{y}, @var{x}, @var{rr}, @var{r}, @var{alt})
Perform an t test for the null hypothesis @code{@var{rr} * @var{b} =
@var{r}} in a classical normal regression model @code{@var{y} =
@var{x} * @var{b} + @var{e}}.  Under the null, the test statistic @var{t}
follows a @var{t} distribution with @var{df} degrees of freedom.

If @var{r} is omitted, a value of 0 is assumed.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{@var{rr} * @var{b} != @var{r}}.  If @var{alt} is @code{">"}, the
one-sided alternative @code{@var{rr} * @var{b} > @var{r}} is used.
Similarly for @var{"<"}, the one-sided alternative @code{@var{rr} *
@var{b} < @var{r}} is used.  The default is the two-sided case. 

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-u_test}
@deftypefn {Function File} {[@var{pval}, @var{z}] =} u_test (@var{x}, @var{y}, @var{alt})
For two samples @var{x} and @var{y}, perform a Mann-Whitney U-test of
the null hypothesis PROB (@var{x} > @var{y}) == 1/2 == PROB (@var{x}
< @var{y}).  Under the null, the test statistic @var{z} approximately
follows a standard normal distribution.  Note that this test is
equivalent to the Wilcoxon rank-sum test.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
PROB (@var{x} > @var{y}) != 1/2.  If @var{alt} is @code{">"}, the
one-sided alternative PROB (@var{x} > @var{y}) > 1/2 is considered.
Similarly for @code{"<"}, the one-sided alternative PROB (@var{x} >
@var{y}) < 1/2 is considered,  The default is the two-sided case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-var_test}
@deftypefn {Function File} {[@var{pval}, @var{f}, @var{df_num}, @var{df_den}] =} var_test (@var{x}, @var{y}, @var{alt})
For two samples @var{x} and @var{y} from normal distributions with
unknown means and unknown variances, perform an F-test of the null
hypothesis of equal variances.  Under the null, the test statistic f
follows an F-distribution with df_num and df_den degrees of freedom.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{var (@var{x}) != var (@var{y})}.  If @var{alt} is @code{">"},
the one-sided alternative @code{var (@var{x}) > var (@var{y})} is
used.  Similarly for "<", the one-sided alternative @code{var
(@var{x}) > var (@var{y})} is used.  The default is the two-sided
case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-welch_test}
@deftypefn {Function File} {[@var{pval}, @var{t}, @var{df}] =} welch_test (@var{x}, @var{y}, @var{alt})
For two samples @var{x} and @var{y} from normal distributions with
unknown means and unknown and not necessarily equal variances,
perform a Welch test of the null hypothesis of equal means.
Under the null, the test statistic t approximately follows a Student
distribution with df degrees of freedom.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{mean (@var{x}) != @var{m}}.  If @var{alt} is @code{">"}, the
one-sided alternative mean(x) > @var{m} is considered.  Similarly for
@code{"<"}, the one-sided alternative mean(x) < @var{m} is
considered.  The default is the two-sided case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-wilcoxon_test}
@deftypefn {Function File} {[@var{pval}, @var{z}] =} wilcoxon_test (@var{x}, @var{y}, @var{alt})
For two matched-pair sample vectors @var{x} and @var{y}, perform a
Wilcoxon signed-rank test of the null hypothesis PROB (@var{x} >
@var{y}) == 1/2.  Under the null, the test statistic @var{z}
approximately follows a standard normal distribution.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
PROB (@var{x} > @var{y}) != 1/2.  If alt is @code{">"}, the one-sided
alternative PROB (@var{x} > @var{y}) > 1/2 is considered.  Similarly
for @code{"<"}, the one-sided alternative PROB (@var{x} > @var{y}) <
1/2 is considered.  The default is the two-sided case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed.
@end deftypefn


@anchor{doc-z_test}
@deftypefn {Function File} {[@var{pval}, @var{z}] =} z_test (@var{x}, @var{m}, @var{v}, @var{alt})
Perform a Z-test of the null hypothesis @code{mean (@var{x}) ==
@var{m}} for a sample @var{x} from a normal distribution with unknown
mean and known variance @var{v}.  Under the null, the test statistic
@var{z} follows a standard normal distribution.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{mean (@var{x}) != @var{m}}.  If @var{alt} is @code{">"}, the
one-sided alternative @code{mean (@var{x}) > @var{m}} is considered.
Similarly for @code{"<"}, the one-sided alternative @code{mean
(@var{x}) < @var{m}} is considered.  The default is the two-sided
case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed
along with some information.
@end deftypefn


@anchor{doc-z_test_2}
@deftypefn {Function File} {[@var{pval}, @var{z}] =} z_test_2 (@var{x}, @var{y}, @var{v_x}, @var{v_y}, @var{alt})
For two samples @var{x} and @var{y} from normal distributions with
unknown means and known variances @var{v_x} and @var{v_y}, perform a
Z-test of the hypothesis of equal means.  Under the null, the test
statistic @var{z} follows a standard normal distribution.

With the optional argument string @var{alt}, the alternative of
interest can be selected.  If @var{alt} is @code{"!="} or
@code{"<>"}, the null is tested against the two-sided alternative
@code{mean (@var{x}) != mean (@var{y})}.  If alt is @code{">"}, the
one-sided alternative @code{mean (@var{x}) > mean (@var{y})} is used.
Similarly for @code{"<"}, the one-sided alternative @code{mean
(@var{x}) < mean (@var{y})} is used.  The default is the two-sided
case.

The p-value of the test is returned in @var{pval}.

If no output argument is given, the p-value of the test is displayed
along with some information.
@end deftypefn


@node Models
@section Models

@anchor{doc-logistic_regression}
@deftypefn {Functio File} {[@var{theta}, @var{beta}, @var{dev}, @var{dl}, @var{d2l}, @var{p}] =} logistic_regression (@var{y}, @var{x}, @var{print}, @var{theta}, @var{beta})
Perform ordinal logistic regression.

Suppose @var{y} takes values in @var{k} ordered categories, and let
@code{gamma_i (@var{x})} be the cumulative probability that @var{y}
falls in one of the first @var{i} categories given the covariate
@var{x}.  Then

@example
[theta, beta] = logistic_regression (y, x)
@end example

@noindent
fits the model

@example
logit (gamma_i (x)) = theta_i - beta' * x,   i = 1, ..., k-1
@end example

The number of ordinal categories, @var{k}, is taken to be the number
of distinct values of @code{round (@var{y})}.  If @var{k} equals 2,
@var{y} is binary and the model is ordinary logistic regression.  The
matrix @var{x} is assumed to have full column rank.

Given @var{y} only, @code{theta = logistic_regression (y)}
fits the model with baseline logit odds only.

The full form is

@example
[theta, beta, dev, dl, d2l, gamma]
   = logistic_regression (y, x, print, theta, beta)
@end example

@noindent
in which all output arguments and all input arguments except @var{y}
are optional.

Stting @var{print} to 1 requests summary information about the fitted
model to be displayed.  Setting @var{print} to 2 requests information
about convergence at each iteration.  Other values request no
information to be displayed.  The input arguments @var{theta} and
@var{beta} give initial estimates for @var{theta} and @var{beta}.

The returned value @var{dev} holds minus twice the log-likelihood.

The returned values @var{dl} and @var{d2l} are the vector of first
and the matrix of second derivatives of the log-likelihood with
respect to @var{theta} and @var{beta}.

@var{p} holds estimates for the conditional distribution of @var{y}
given @var{x}.
@end deftypefn


@node Distributions
@section Distributions

@anchor{doc-beta_cdf}
@deftypefn {Function File} {} beta_cdf (@var{x}, @var{a}, @var{b})
For each element of @var{x}, returns the CDF at @var{x} of the beta
distribution with parameters @var{a} and @var{b}, i.e.,
PROB (beta (@var{a}, @var{b}) <= @var{x}).
@end deftypefn


@anchor{doc-beta_inv}
@deftypefn {Function File} {} beta_inv (@var{x}, @var{a}, @var{b})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the Beta distribution with parameters @var{a}
and @var{b}.
@end deftypefn


@anchor{doc-beta_pdf}
@deftypefn {Function File} {} beta_pdf (@var{x}, @var{a}, @var{b})
For each element of @var{x}, returns the PDF at @var{x} of the beta
distribution with parameters @var{a} and @var{b}.
@end deftypefn


@anchor{doc-beta_rnd}
@deftypefn {Function File} {} beta_rnd (@var{a}, @var{b}, @var{r}, @var{c})
@deftypefnx {Function File} {} beta_rnd (@var{a}, @var{b}, @var{sz})
Return an @var{r} by @var{c} or @code{size (@var{sz})} matrix of 
random samples from the Beta distribution with parameters @var{a} and
@var{b}.  Both @var{a} and @var{b} must be scalar or of size @var{r}
 by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{a} and @var{b}.
@end deftypefn


@anchor{doc-binomial_cdf}
@deftypefn {Function File} {} binomial_cdf (@var{x}, @var{n}, @var{p})
For each element of @var{x}, compute the CDF at @var{x} of the
binomial distribution with parameters @var{n} and @var{p}.
@end deftypefn


@anchor{doc-binomial_inv}
@deftypefn {Function File} {} binomial_inv (@var{x}, @var{n}, @var{p})
For each element of @var{x}, compute the quantile at @var{x} of the
binomial distribution with parameters @var{n} and @var{p}.
@end deftypefn


@anchor{doc-binomial_pdf}
@deftypefn {Function File} {} binomial_pdf (@var{x}, @var{n}, @var{p})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the binomial distribution with parameters @var{n}
and @var{p}.
@end deftypefn


@anchor{doc-binomial_rnd}
@deftypefn {Function File} {} binomial_rnd (@var{n}, @var{p}, @var{r}, @var{c})
@deftypefnx {Function File} {} binomial_rnd (@var{n}, @var{p}, @var{sz})
Return an @var{r} by @var{c}  or a @code{size (@var{sz})} matrix of 
random samples from the binomial distribution with parameters @var{n}
and @var{p}.  Both @var{n} and @var{p} must be scalar or of size
@var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{n} and @var{p}.
@end deftypefn


@anchor{doc-cauchy_cdf}
@deftypefn {Function File} {} cauchy_cdf (@var{x}, @var{lambda}, @var{sigma})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the Cauchy distribution with location
parameter @var{lambda} and scale parameter @var{sigma}.  Default
values are @var{lambda} = 0, @var{sigma} = 1. 
@end deftypefn


@anchor{doc-cauchy_inv}
@deftypefn {Function File} {} cauchy_inv (@var{x}, @var{lambda}, @var{sigma})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the Cauchy distribution with location parameter
@var{lambda} and scale parameter @var{sigma}.  Default values are
@var{lambda} = 0, @var{sigma} = 1. 
@end deftypefn


@anchor{doc-cauchy_pdf}
@deftypefn {Function File} {} cauchy_pdf (@var{x}, @var{lambda}, @var{sigma})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the Cauchy distribution with location parameter
@var{lambda} and scale parameter @var{sigma} > 0.  Default values are
@var{lambda} = 0, @var{sigma} = 1. 
@end deftypefn


@anchor{doc-cauchy_rnd}
@deftypefn {Function File} {} cauchy_rnd (@var{lambda}, @var{sigma}, @var{r}, @var{c})
@deftypefnx {Function File} {} cauchy_rnd (@var{lambda}, @var{sigma}, @var{sz})
Return an @var{r} by @var{c} or a @code{size (@var{sz})} matrix of 
random samples from the Cauchy distribution with parameters @var{lambda} 
and @var{sigma} which must both be scalar or of size @var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{lambda} and @var{sigma}.
@end deftypefn


@anchor{doc-chisquare_cdf}
@deftypefn {Function File} {} chisquare_cdf (@var{x}, @var{n})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the chisquare distribution with @var{n}
degrees of freedom.
@end deftypefn


@anchor{doc-chisquare_inv}
@deftypefn {Function File} {} chisquare_inv (@var{x}, @var{n})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the chisquare distribution with @var{n} degrees of
freedom.
@end deftypefn


@anchor{doc-chisquare_pdf}
@deftypefn {Function File} {} chisquare_pdf (@var{x}, @var{n})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the chisquare distribution with @var{k} degrees
of freedom.
@end deftypefn


@anchor{doc-chisquare_rnd}
@deftypefn {Function File} {} chisquare_rnd (@var{n}, @var{r}, @var{c})
@deftypefnx {Function File} {} chisquare_rnd (@var{n}, @var{sz})
Return an @var{r} by @var{c}  or a @code{size (@var{sz})} matrix of 
random samples from the chisquare distribution with @var{n} degrees 
of freedom.  @var{n} must be a scalar or of size @var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the size of @var{n}.
@end deftypefn


@anchor{doc-discrete_cdf}
@deftypefn {Function File} {} discrete_cdf (@var{x}, @var{v}, @var{p})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of a univariate discrete distribution which
assumes the values in v with probabilities @var{p}.
@end deftypefn


@anchor{doc-discrete_inv}
@deftypefn {Function File} {} discrete_inv (@var{x}, @var{v}, @var{p})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the univariate distribution which assumes the
values in @var{v} with probabilities @var{p}.
@end deftypefn


@anchor{doc-discrete_pdf}
@deftypefn {Function File} {} discrete_pdf (@var{x}, @var{v}, @var{p})
For each element of @var{x}, compute the probability density function
(pDF) at @var{x} of a univariate discrete distribution which assumes
the values in @var{v} with probabilities @var{p}.
@end deftypefn


@anchor{doc-discrete_rnd}
@deftypefn {Function File} {} discrete_rnd (@var{n}, @var{v}, @var{p})
@deftypefnx {Function File} {} discrete_rnd (@var{v}, @var{p}, @var{r}, @var{c})
@deftypefnx {Function File} {} discrete_rnd (@var{v}, @var{p}, @var{sz})
Generate a row vector containing a random sample of size @var{n} from
the univariate distribution which assumes the values in @var{v} with
probabilities @var{p}. @var{n} must be a scalar.

If @var{r} and @var{c} are given create a matrix with @var{r} rows and
@var{c} columns. Or if @var{sz} is a vector, create a matrix of size
@var{sz}.
@end deftypefn


@anchor{doc-empirical_cdf}
@deftypefn {Function File} {} empirical_cdf (@var{x}, @var{data})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the empirical distribution obtained from
the univariate sample @var{data}.
@end deftypefn


@anchor{doc-empirical_inv}
@deftypefn {Function File} {} empirical_inv (@var{x}, @var{data})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the empirical distribution obtained from the
univariate sample @var{data}.
@end deftypefn


@anchor{doc-empirical_pdf}
@deftypefn {Function File} {} empirical_pdf (@var{x}, @var{data})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the empirical distribution obtained from the
univariate sample @var{data}.
@end deftypefn


@anchor{doc-empirical_rnd}
@deftypefn {Function File} {} empirical_rnd (@var{n}, @var{data})
@deftypefnx {Function File} {} empirical_rnd (@var{data}, @var{r}, @var{c})
@deftypefnx {Function File} {} empirical_rnd (@var{data}, @var{sz})
Generate a bootstrap sample of size @var{n} from the empirical
distribution obtained from the univariate sample @var{data}.

If @var{r} and @var{c} are given create a matrix with @var{r} rows and
@var{c} columns. Or if @var{sz} is a vector, create a matrix of size
@var{sz}.
@end deftypefn


@anchor{doc-exponential_cdf}
@deftypefn {Function File} {} exponential_cdf (@var{x}, @var{lambda})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the exponential distribution with
parameter @var{lambda}.

The arguments can be of common size or scalar.
@end deftypefn


@anchor{doc-exponential_inv}
@deftypefn {Function File} {} exponential_inv (@var{x}, @var{lambda})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the exponential distribution with parameter
@var{lambda}.
@end deftypefn


@anchor{doc-exponential_pdf}
@deftypefn {Function File} {} exponential_pdf (@var{x}, @var{lambda})
For each element of @var{x}, compute the probability density function
(PDF) of the exponential distribution with parameter @var{lambda}.
@end deftypefn


@anchor{doc-exponential_rnd}
@deftypefn {Function File} {} exponential_rnd (@var{lambda}, @var{r}, @var{c})
@deftypefnx {Function File} {} exponential_rnd (@var{lambda}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the
exponential distribution with parameter @var{lambda}, which must be a
scalar or of size @var{r} by @var{c}. Or if @var{sz} is a vector, 
create a matrix of size @var{sz}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the size of @var{lambda}.
@end deftypefn


@anchor{doc-f_cdf}
@deftypefn {Function File} {} f_cdf (@var{x}, @var{m}, @var{n})
For each element of @var{x}, compute the CDF at @var{x} of the F
distribution with @var{m} and @var{n} degrees of freedom, i.e.,
PROB (F (@var{m}, @var{n}) <= @var{x}). 
@end deftypefn


@anchor{doc-f_inv}
@deftypefn {Function File} {} f_inv (@var{x}, @var{m}, @var{n})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the F distribution with parameters @var{m} and
@var{n}.
@end deftypefn


@anchor{doc-f_pdf}
@deftypefn {Function File} {} f_pdf (@var{x}, @var{m}, @var{n})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the F distribution with @var{m} and @var{n}
degrees of freedom.
@end deftypefn


@anchor{doc-f_rnd}
@deftypefn {Function File} {} f_rnd (@var{m}, @var{n}, @var{r}, @var{c})
@deftypefnx {Function File} {} f_rnd (@var{m}, @var{n}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the F
distribution with @var{m} and @var{n} degrees of freedom.  Both
@var{m} and @var{n} must be scalar or of size @var{r} by @var{c}.
If @var{sz} is a vector the random samples are in a matrix of 
size @var{sz}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{m} and @var{n}.
@end deftypefn


@anchor{doc-gamma_cdf}
@deftypefn {Function File} {} gamma_cdf (@var{x}, @var{a}, @var{b})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the Gamma distribution with parameters
@var{a} and @var{b}.
@end deftypefn


@anchor{doc-gamma_inv}
@deftypefn {Function File} {} gamma_inv (@var{x}, @var{a}, @var{b})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the Gamma distribution with parameters @var{a}
and @var{b}. 
@end deftypefn


@anchor{doc-gamma_pdf}
@deftypefn {Function File} {} gamma_pdf (@var{x}, @var{a}, @var{b})
For each element of @var{x}, return the probability density function
(PDF) at @var{x} of the Gamma distribution with parameters @var{a}
and @var{b}.
@end deftypefn


@anchor{doc-gamma_rnd}
@deftypefn {Function File} {} gamma_rnd (@var{a}, @var{b}, @var{r}, @var{c})
@deftypefnx {Function File} {} gamma_rnd (@var{a}, @var{b}, @var{sz})
Return an @var{r} by @var{c} or a @code{size (@var{sz})} matrix of 
random samples from the Gamma distribution with parameters @var{a}
and @var{b}.  Both @var{a} and @var{b} must be scalar or of size 
@var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{a} and @var{b}.
@end deftypefn


@anchor{doc-geometric_cdf}
@deftypefn {Function File} {} geometric_cdf (@var{x}, @var{p})
For each element of @var{x}, compute the CDF at @var{x} of the
geometric distribution with parameter @var{p}.
@end deftypefn


@anchor{doc-geometric_inv}
@deftypefn {Function File} {} geometric_inv (@var{x}, @var{p})
For each element of @var{x}, compute the quantile at @var{x} of the
geometric distribution with parameter @var{p}.
@end deftypefn


@anchor{doc-geometric_pdf}
@deftypefn {Function File} {} geometric_pdf (@var{x}, @var{p})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the geometric distribution with parameter @var{p}.
@end deftypefn


@anchor{doc-geometric_rnd}
@deftypefn {Function File} {} geometric_rnd (@var{p}, @var{r}, @var{c})
@deftypefnx {Function File} {} geometric_rnd (@var{p}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the
geometric distribution with parameter @var{p}, which must be a scalar
or of size @var{r} by @var{c}.

If @var{r} and @var{c} are given create a matrix with @var{r} rows and
@var{c} columns. Or if @var{sz} is a vector, create a matrix of size
@var{sz}.
@end deftypefn


@anchor{doc-hypergeometric_cdf}
@deftypefn {Function File} {} hypergeometric_cdf (@var{x}, @var{m}, @var{t}, @var{n})
Compute the cumulative distribution function (CDF) at @var{x} of the
hypergeometric distribution with parameters @var{m}, @var{t}, and
@var{n}.  This is the probability of obtaining not more than @var{x}
marked items when randomly drawing a sample of size @var{n} without
replacement from a population of total size @var{t} containing
@var{m} marked items.

The parameters @var{m}, @var{t}, and @var{n} must positive integers
with @var{m} and @var{n} not greater than @var{t}.
@end deftypefn


@anchor{doc-hypergeometric_inv}
@deftypefn {Function File} {} hypergeometric_inv (@var{x}, @var{m}, @var{t}, @var{n})
For each element of @var{x}, compute the quantile at @var{x} of the
hypergeometric distribution with parameters @var{m}, @var{t}, and
@var{n}.

The parameters @var{m}, @var{t}, and @var{n} must positive integers
with @var{m} and @var{n} not greater than @var{t}.
@end deftypefn


@anchor{doc-hypergeometric_pdf}
@deftypefn {Function File} {} hypergeometric_pdf (@var{x}, @var{m}, @var{t}, @var{n})
Compute the probability density function (PDF) at @var{x} of the
hypergeometric distribution with parameters @var{m}, @var{t}, and
@var{n}. This is the probability of obtaining @var{x} marked items
when randomly drawing a sample of size @var{n} without replacement
from a population of total size @var{t} containing @var{m} marked items.

The arguments must be of common size or scalar.
@end deftypefn


@anchor{doc-hypergeometric_rnd}
@deftypefn {Function File} {} hypergeometric_rnd (@var{n_size}, @var{m}, @var{t}, @var{n})
@deftypefnx {Function File} {} hypergeometric_rnd (@var{m}, @var{t}, @var{n}, @var{r}, @var{c})
@deftypefnx {Function File} {} hypergeometric_rnd (@var{m}, @var{t}, @var{n}, @var{sz})
Generate a row vector containing a random sample of size @var{n_size}
from the hypergeometric distribution with parameters @var{m}, @var{t},
and @var{n}.

If  @var{r} and @var{c} are given create a matrix with @var{r} rows and
@var{c} columns. Or if @var{sz} is a vector, create a matrix of size
@var{sz}.

The parameters @var{m}, @var{t}, and @var{n} must positive integers
with @var{m} and @var{n} not greater than @var{t}.
@end deftypefn


@anchor{doc-kolmogorov_smirnov_cdf}
@deftypefn {Function File} {} kolmogorov_smirnov_cdf (@var{x}, @var{tol})
Return the CDF at @var{x} of the Kolmogorov-Smirnov distribution,
@iftex
@tex
$$ Q(x) = sum_{k=-\infty}^\infty (-1)^k exp(-2 k^2 x^2) $$
@end tex
@end iftex
@ifinfo
@example
         Inf
Q(x) =   SUM    (-1)^k exp(-2 k^2 x^2)
       k = -Inf
@end example
@end ifinfo

@noindent
for @var{x} > 0.

The optional parameter @var{tol} specifies the precision up to which
the series should be evaluated;  the default is @var{tol} = @code{eps}.
@end deftypefn


@anchor{doc-laplace_cdf}
@deftypefn {Function File} {} laplace_cdf (@var{x})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the Laplace distribution.
@end deftypefn


@anchor{doc-laplace_inv}
@deftypefn {Function File} {} laplace_inv (@var{x})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the Laplace distribution.
@end deftypefn


@anchor{doc-laplace_pdf}
@deftypefn {Function File} {} laplace_pdf (@var{x})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the Laplace distribution.
@end deftypefn


@anchor{doc-laplace_rnd}
@deftypefn {Function File} {} laplace_rnd (@var{r}, @var{c})
@deftypefnx {Function File} {} laplace_rnd (@var{sz});
Return an @var{r} by @var{c} matrix of random numbers from the
Laplace distribution. Or is @var{sz} is a vector, create a matrix of
@var{sz}.
@end deftypefn


@anchor{doc-logistic_cdf}
@deftypefn {Function File} {} logistic_cdf (@var{x})
For each component of @var{x}, compute the CDF at @var{x} of the
logistic distribution.
@end deftypefn


@anchor{doc-logistic_inv}
@deftypefn {Function File} {} logistic_inv (@var{x})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the logistic distribution.
@end deftypefn


@anchor{doc-logistic_pdf}
@deftypefn {Function File} {} logistic_pdf (@var{x})
For each component of @var{x}, compute the PDF at @var{x} of the
logistic distribution.
@end deftypefn


@anchor{doc-logistic_rnd}
@deftypefn {Function File} {} logistic_rnd (@var{r}, @var{c})
@deftypefnx {Function File} {} logistic_rnd (@var{sz})
Return an @var{r} by @var{c} matrix of random numbers from the
logistic distribution. Or is @var{sz} is a vector, create a matrix of
@var{sz}.
@end deftypefn


@anchor{doc-lognormal_cdf}
@deftypefn {Function File} {} lognormal_cdf (@var{x}, @var{a}, @var{v})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the lognormal distribution with
parameters @var{a} and @var{v}.  If a random variable follows this
distribution, its logarithm is normally distributed with mean
@code{log (@var{a})} and variance @var{v}.

Default values are @var{a} = 1, @var{v} = 1.
@end deftypefn


@anchor{doc-lognormal_inv}
@deftypefn {Function File} {} lognormal_inv (@var{x}, @var{a}, @var{v})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the lognormal distribution with parameters @var{a}
and @var{v}.  If a random variable follows this distribution, its
logarithm is normally distributed with mean @code{log (@var{a})} and
variance @var{v}.

Default values are @var{a} = 1, @var{v} = 1.
@end deftypefn


@anchor{doc-lognormal_pdf}
@deftypefn {Function File} {} lognormal_pdf (@var{x}, @var{a}, @var{v})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the lognormal distribution with parameters
@var{a} and @var{v}.  If a random variable follows this distribution,
its logarithm is normally distributed with mean @code{log (@var{a})}
and variance @var{v}.

Default values are @var{a} = 1, @var{v} = 1.
@end deftypefn


@anchor{doc-lognormal_rnd}
@deftypefn {Function File} {} lognormal_rnd (@var{a}, @var{v}, @var{r}, @var{c})
@deftypefnx {Function File} {} lognormal_rnd (@var{a}, @var{v}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the
lognormal distribution with parameters @var{a} and @var{v}. Both
@var{a} and @var{v} must be scalar or of size @var{r} by @var{c}.
Or if @var{sz} is a vector, create a matrix of size @var{sz}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{a} and @var{v}.
@end deftypefn


@anchor{doc-normal_cdf}
@deftypefn {Function File} {} normal_cdf (@var{x}, @var{m}, @var{v})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the normal distribution with mean
@var{m} and variance @var{v}.

Default values are @var{m} = 0, @var{v} = 1.
@end deftypefn


@anchor{doc-normal_inv}
@deftypefn {Function File} {} normal_inv (@var{x}, @var{m}, @var{v})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the normal distribution with mean @var{m} and
variance @var{v}.

Default values are @var{m} = 0, @var{v} = 1.
@end deftypefn


@anchor{doc-normal_pdf}
@deftypefn {Function File} {} normal_pdf (@var{x}, @var{m}, @var{v})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the normal distribution with mean @var{m} and
variance @var{v}.

Default values are @var{m} = 0, @var{v} = 1.
@end deftypefn


@anchor{doc-normal_rnd}
@deftypefn {Function File} {} normal_rnd (@var{m}, @var{v}, @var{r}, @var{c})
@deftypefnx {Function File} {} normal_rnd (@var{m}, @var{v}, @var{sz})
Return an @var{r} by @var{c}  or @code{size (@var{sz})} matrix of
random samples from the normal distribution with parameters @var{m} 
and @var{v}.  Both @var{m} and @var{v} must be scalar or of size 
@var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{m} and @var{v}.
@end deftypefn


@anchor{doc-pascal_cdf}
@deftypefn {Function File} {} pascal_cdf (@var{x}, @var{n}, @var{p})
For each element of @var{x}, compute the CDF at x of the Pascal
(negative binomial) distribution with parameters @var{n} and @var{p}.

The number of failures in a Bernoulli experiment with success
probability @var{p} before the @var{n}-th success follows this
distribution.
@end deftypefn


@anchor{doc-pascal_inv}
@deftypefn {Function File} {} pascal_inv (@var{x}, @var{n}, @var{p})
For each element of @var{x}, compute the quantile at @var{x} of the
Pascal (negative binomial) distribution with parameters @var{n} and
@var{p}.

The number of failures in a Bernoulli experiment with success
probability @var{p} before the @var{n}-th success follows this
distribution.
@end deftypefn


@anchor{doc-pascal_pdf}
@deftypefn {Function File} {} pascal_pdf (@var{x}, @var{n}, @var{p})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the Pascal (negative binomial) distribution with
parameters @var{n} and @var{p}.

The number of failures in a Bernoulli experiment with success
probability @var{p} before the @var{n}-th success follows this
distribution. 
@end deftypefn


@anchor{doc-pascal_rnd}
@deftypefn {Function File} {} pascal_rnd (@var{n}, @var{p}, @var{r}, @var{c})
@deftypefnx {Function File} {} pascal_rnd (@var{n}, @var{p}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the Pascal
(negative binomial) distribution with parameters @var{n} and @var{p}.
Both @var{n} and @var{p} must be scalar or of size @var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{n} and @var{p}. Or if @var{sz} is a vector, 
create a matrix of size @var{sz}.
@end deftypefn


@anchor{doc-poisson_cdf}
@deftypefn {Function File} {} poisson_cdf (@var{x}, @var{lambda})
For each element of @var{x}, compute the cumulative distribution
function (CDF) at @var{x} of the Poisson distribution with parameter
lambda.
@end deftypefn


@anchor{doc-poisson_inv}
@deftypefn {Function File} {} poisson_inv (@var{x}, @var{lambda})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the Poisson distribution with parameter
@var{lambda}.
@end deftypefn


@anchor{doc-poisson_pdf}
@deftypefn {Function File} {} poisson_pdf (@var{x}, @var{lambda})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the poisson distribution with parameter @var{lambda}.
@end deftypefn


@anchor{doc-poisson_rnd}
@deftypefn {Function File} {} poisson_rnd (@var{lambda}, @var{r}, @var{c})
Return an @var{r} by @var{c} matrix of random samples from the
Poisson distribution with parameter @var{lambda}, which must be a 
scalar or of size @var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the size of @var{lambda}.
@end deftypefn


@anchor{doc-stdnormal_cdf}
@deftypefn {Function File} {} stdnormal_cdf (@var{x})
For each component of @var{x}, compute the CDF of the standard normal
distribution at @var{x}.
@end deftypefn


@anchor{doc-stdnormal_inv}
@deftypefn {Function File} {} stdnormal_inv (@var{x})
For each component of @var{x}, compute compute the quantile (the
inverse of the CDF) at @var{x} of the standard normal distribution.
@end deftypefn


@anchor{doc-stdnormal_pdf}
@deftypefn {Function File} {} stdnormal_pdf (@var{x})
For each element of @var{x}, compute the probability density function
(PDF) of the standard normal distribution at @var{x}.
@end deftypefn


@anchor{doc-stdnormal_rnd}
@deftypefn {Function File} {} stdnormal_rnd (@var{r}, @var{c})
@deftypefnx {Function File} {} stdnormal_rnd (@var{sz})
Return an @var{r} by @var{c} or @code{size (@var{sz})} matrix of 
random numbers from the standard normal distribution.
@end deftypefn


@anchor{doc-t_cdf}
@deftypefn {Function File} {} t_cdf (@var{x}, @var{n})
For each element of @var{x}, compute the CDF at @var{x} of the
t (Student) distribution with @var{n} degrees of freedom, i.e.,
PROB (t(@var{n}) <= @var{x}).
@end deftypefn


@anchor{doc-t_inv}
@deftypefn {Function File} {} t_inv (@var{x}, @var{n})
For each component of @var{x}, compute the quantile (the inverse of
the CDF) at @var{x} of the t (Student) distribution with parameter
@var{n}.
@end deftypefn


@anchor{doc-t_pdf}
@deftypefn {Function File} {} t_pdf (@var{x}, @var{n})
For each element of @var{x}, compute the probability density function
(PDF) at @var{x} of the @var{t} (Student) distribution with @var{n}
degrees of freedom. 
@end deftypefn


@anchor{doc-t_rnd}
@deftypefn {Function File} {} t_rnd (@var{n}, @var{r}, @var{c})
@deftypefnx {Function File} {} t_rnd (@var{n}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the t
(Student) distribution with @var{n} degrees of freedom.  @var{n} must
be a scalar or of size @var{r} by @var{c}. Or if @var{sz} is a
vector create a matrix of size @var{sz}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the size of @var{n}.
@end deftypefn


@anchor{doc-uniform_cdf}
@deftypefn {Function File} {} uniform_cdf (@var{x}, @var{a}, @var{b})
Return the CDF at @var{x} of the uniform distribution on [@var{a},
@var{b}], i.e., PROB (uniform (@var{a}, @var{b}) <= x).

Default values are @var{a} = 0, @var{b} = 1.
@end deftypefn


@anchor{doc-uniform_inv}
@deftypefn {Function File} {} uniform_inv (@var{x}, @var{a}, @var{b})
For each element of @var{x}, compute the quantile (the inverse of the
CDF) at @var{x} of the uniform distribution on [@var{a}, @var{b}].

Default values are @var{a} = 0, @var{b} = 1.
@end deftypefn


@anchor{doc-uniform_pdf}
@deftypefn {Function File} {} uniform_pdf (@var{x}, @var{a}, @var{b})
For each element of @var{x}, compute the PDF at @var{x} of the uniform
distribution on [@var{a}, @var{b}].

Default values are @var{a} = 0, @var{b} = 1.
@end deftypefn


@anchor{doc-uniform_rnd}
@deftypefn {Function File} {} uniform_rnd (@var{a}, @var{b}, @var{r}, @var{c})
@deftypefnx {Function File} {} uniform_rnd (@var{a}, @var{b}, @var{sz})
Return an @var{r} by @var{c} or a @code{size (@var{sz})} matrix of 
random samples from the uniform distribution on [@var{a}, @var{b}]. 
Both @var{a} and @var{b} must be scalar or of size @var{r} by @var{c}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{a} and @var{b}.
@end deftypefn


@anchor{doc-weibull_cdf}
@deftypefn {Function File} {} weibull_cdf (@var{x}, @var{alpha}, @var{sigma})
Compute the cumulative distribution function (CDF) at @var{x} of the
Weibull distribution with shape parameter @var{alpha} and scale
parameter @var{sigma}, which is

@example
1 - exp(-(x/sigma)^alpha)
@end example

@noindent
for @var{x} >= 0.
@end deftypefn


@anchor{doc-weibull_inv}
@deftypefn {Function File} {} weibull_inv (@var{x}, @var{lambda}, @var{alpha})
Compute the quantile (the inverse of the CDF) at @var{x} of the
Weibull distribution with shape parameter @var{alpha} and scale
parameter @var{sigma}.
@end deftypefn


@anchor{doc-weibull_pdf}
@deftypefn {Function File} {} weibull_pdf (@var{x}, @var{alpha}, @var{sigma})
Compute the probability density function (PDF) at @var{x} of the
Weibull distribution with shape parameter @var{alpha} and scale
parameter @var{sigma} which is given by

@example
   alpha * sigma^(-alpha) * x^(alpha-1) * exp(-(x/sigma)^alpha)
@end example

@noindent
for @var{x} > 0.
@end deftypefn


@anchor{doc-weibull_rnd}
@deftypefn {Function File} {} weibull_rnd (@var{alpha}, @var{sigma}, @var{r}, @var{c})
@deftypefnx {Function File} {} weibull_rnd (@var{alpha}, @var{sigma}, @var{sz})
Return an @var{r} by @var{c} matrix of random samples from the
Weibull distribution with parameters @var{alpha} and @var{sigma}
which must be scalar or of size @var{r} by @var{c}. Or if @var{sz}
is a vector return a matrix of size @var{sz}.

If @var{r} and @var{c} are omitted, the size of the result matrix is
the common size of @var{alpha} and @var{sigma}.
@end deftypefn


@anchor{doc-wiener_rnd}
@deftypefn {Function File} {} wiener_rnd (@var{t}, @var{d}, @var{n})
Return a simulated realization of the @var{d}-dimensional Wiener Process
on the interval [0, @var{t}].  If @var{d} is omitted, @var{d} = 1 is
used. The first column of the return matrix contains time, the
remaining columns contain the Wiener process.

The optional parameter @var{n} gives the number of summands used for
simulating the process over an interval of length 1.  If @var{n} is
omitted, @var{n} = 1000 is used.
@end deftypefn

